Find $\cos\left(165^\circ\right)$ exactly using an angle addition or subtraction formula. Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{1+\sqrt{3}}{2}$ (Choice B) B $\dfrac{-1-\sqrt{3}}{2}$ (Choice C) C $\dfrac{\sqrt{8}}{3}$ (Choice D) D $\dfrac{-\sqrt{2}-\sqrt{6}}{4}$
Answer: The strategy First, we should rewrite the given angle $165^\circ$ as the sum or difference of two special angles. Then, we can use the cosine addition or subtraction identities in order to evaluate $\cos\left(165^\circ\right)$. [How do we find the trigonometric value of a sum or difference?] Rewriting $165^\circ$ We can rewrite $165^\circ$ as follows. $\begin{aligned}165^\circ&=225^\circ-60^\circ\end{aligned}$ In other words, $165^\circ$ is the difference of the special angles $225^\circ$ and $60^\circ$. Evaluating $\cos\left(165^\circ\right)$ Using the cosine subtraction identity, we get the following. $\begin{aligned} \cos\left(165^\circ\right)&= \cos\left(225^\circ-60^\circ\right) \\\\\\ &= \cos \left(225^\circ\right) \cos \left(60^\circ\right) + \sin \left(225^\circ\right) \sin \left(60^\circ\right) \\\\\\ &=\left(-\dfrac{\sqrt{2}}{2}\right) \left(\dfrac{1}{2}\right) + \left(-\dfrac{\sqrt{2}}{2}\right) \left(\dfrac{\sqrt{3}}{2}\right) \\\\\\ &=\left(-\dfrac{\sqrt{2}}{4}\right) + \left(-\dfrac{\sqrt{6}}{4}\right)\\\\\\ &=\dfrac{-\sqrt{2}-\sqrt{6}}{4} \end{aligned}$ Summary $\cos\left(165^\circ\right) = \dfrac{-\sqrt{2}-\sqrt{6}}{4}$